Could a person see an object moving away from themselves at lightspeed?
Yes. we can see the images of the object where it was. Now the object is far more away.
The only thing that can move away from you at lightspeed is light itself and once it leaves your vicinity, you can't see it unless it hits something and reflects back to you. Similarly, an object moving at near lightspeed would not be visible unless a light was shining on it and reflecting light back to you or unless it was giving off its own light. But in these cases the light would be severely red-shifted into the invisible range. Then you also have the problem of being able to see a fast moving object that wasn't any where near lightspeed. So I would say it is extremely unlikely that you could see an object moving at near lightspeed away from you.
No object can move at the speed of light, so as phrased the answer is "no, because nothing like that ever has or will exist, and of course you can't see something that doesn't exist".
However, there is a closely related related question that does have a helpful answer: Could a person see an object moving away from themselves a speed of c(1-ε), where ε is an arbitrarily small but non-zero number?
And the answer is yes, although the light emitted from the object may be so redshifted that it's not visible to the naked eye and you'd need special equipment to detect that it's there. (Now, if the object were emitting gamma radiation... Then the redshift might get it down to visible wavelengths...). Of course by the time you see the object it won't be there any more.
Consider a laser beam shining in a theater. If there is some particulate matter in the air like vapor, you can easily see the beam even though it is traveling at lightspeed. But that is because it is on all the time. What if they flashed a short pulse of light, say for one nanosecond. This would produce a beam segment that was about a foot long. Even if it was bright enough, do you think you could watch it as it propagates from the laser to the wall, say 1000 feet away? It would get there in one microsecond. If you had some high speed photographic equipment, it would record that the flash took two microseconds to get to the wall and it would appear to be traveling at one-half the speed of light. But do you think you could observe the propagation of the flash with your naked eye? I think not. If the flash was bright enough for you to see, it would appear simply as a pencil thin laser beam turning on and off very quickly without any hint that it was going from the laser to the wall. This is assuming that you are able to watch the beam traverse across your field of view instead of propagating directly away from you.
Now let's consider a laser flash going from the earth to the moon where it would take about 2.5 seconds from the time the flash was emitted by the laser until the image of the flash hitting the moon reflects back to earth. Let's assume there was some particulate matter between the earth and the moon.
Since the path of the laser light is in line with the observer, although the light pulse would be visible, it would be very difficult to distinguish its position as traveling away but again it would appear just as a spot of light for 2.5 seconds.
My main point is that although the light may be visible, it may be very difficult to see it as traveling away.
OK, now that we have analyzed this, it occurs to me that this is not what lightCurious is concerned about. I'm guessing that he's wondering if the light can even reach an observer since the object is moving away at lightspeed and light itself is moving at lightspeed, will the light simply just not move? Kind of like if we consider a plane traveling at 600 miles per hour, shooting a gun at us with bullets that travel 600 miles per hour, they just won't be going fast enough to hit us. If this is indeed what lightCurious is wondering, the answer is that the light will reach us. It's not like bullets whose speed is subtracted by the speed of the traveling gun. With light, it doesn't matter how fast the source of light is moving, the speed with which the light travels toward us is not affected by the speed of the source. However, it will still be red-shifted.
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